Position of the dividing surface

From both the time-dependent plot and the time-independent projection, it is clear that the transition path crosses the space-fixed dividing surface qu = 0 several times. These crossings are indicated by thick green dots. As expected, therefore, the fixed surface is not free of recrossings and thus does not satisfy the fundamental requirement for an exact TST dividing surface. The moving surface, by contrast, is crossed only once, at the reaction time head = 8.936 that is marked by the blue cut. The solid blue line in this cut shows the instantaneous position of the dividing surface dotted lines indicate coordinate axes. 

The overestimation of the TST rate constant leads to a variational principle for the optimization of the position of the dividing surface constituting the transition state. In general, one can write ... 

In Fig. 5.1.2 we have made a sketch with a few examples of trajectories to illustrate how the position of the dividing surface may influence the results. 

The approach described above will, in general, not give the exact rate constant, since it is based on a quite arbitrary choice of the dividing surface we do not know if the choice is valid according to the Wigner theorem, namely that the rate constant is at a minimum with respect to variations in the choice of dividing surface. A variational determination of the rate constant with respect to the position of the dividing surface is usually not done directly. 

Here II = yo - y is the surface pressure, k the Boltzmann constant, and (Oo the solvent molecule molecular area. An important feature of Eq. (6) is that it involves solvent characteristics only. The (Oo value depends on the choice of the position of the dividing surface. Assuming the solvent adsorption to be positive, the equation was proposed30 which relates the surface excesses H of the solvent (subscript i = 0) and dissolved species (i > 1) with any molecular area (fy ... 

The definition of a invariant with respect to positioning of the dividing surface can be worked out, if one analyzes trends in the/z)-pc(z) function within the discontinuity surface. The specified quantity has the same value in the bulk of both phases, equal to the negative external pressure (Fig. 1-4). Within the discontinuity surface, pressure p has a tensor nature, making Pascal s law invalid. Meanwhile, the concentration and pressure dependence of the surface energy density,/ given by eq. (1.1), is valid only in the regions where Pascal s law holds, i.e., where pressure is a scalar quantity (direct summation of a scalar and a tensor within the same equation is not permitted). 

It is noteworthy that the above treatment is only valid for flat interfaces. Things get more complicated if one deals with curved surfaces, for which it is necessary to consider a pressure gradient existing between two phases in contact. In such a case the surface tension becomes dependent on the position of the dividing surface. A position of the dividing surface that yields a minimum value of a is referred to as the position of the surface of tension , according to Gibbs. 

The excess (per unit area) of internal energy, e, and entropy, rj, within the interfacial layer can be introduced by analogy with the excess of free energy [6]. These quantities are also dependent on the position of the dividing surface. One can verily that the equations relating o, 8, and r are very similar to those derived in conventional three-dimensional thermodynamics, i.e. ... 

According to Gibbs, it is possible to chose a position of the dividing surface such that So = 0, the so-called surface of tension, for which one can write... 

Strictly speaking, such position of the dividing surface differs from that of the equimolecular one (with respect to solvent), however, the difference between the two is too small to cause any significant influence on the results of the present treatment... 

Prove that relative surface excesses are independent of the position of the dividing surface used in the reference system chosen for the thermodynamic treatment of the interface. 

Up to this point we have not specified the position of the dividing surface. If we now choose this so that T, =0, then... 

The treatment of surface phases presented here is essentially that of Gibbs. (The reader is referred to the book J. Willard Gibbs, Collected Works, vol. I, pp. 219-328, Yale University Press, New Haven, Conn., 1948, for more details.) In Sec. 10-1, we derive the fundamental equations for the thermodynamic treatment of surface phenomena. In Sec. 10-2, we consider the dependence of the various surface properties on the position of the dividing surface. Section 10-3 is devoted to a study of the temperature and component derivatives of the surface tension. 

It is sometimes convenient to choose the position of the dividing surface so that one of the quantities E S , or T, is zero. This can be done except when the corresponding densities in the two phases are identical. For example, we may choose the dividing surface to be such that Ti vanishes. In this case, Eq. (10-35) becomes... 

If the condition (2.23) does not hold, then the adsorption isotherms are thermodynamically inconsistent. It should be noted that Eq. (2.23) was derived for the Gibbs dividing surface, located such that the adsorption of the solvent is zero (i=0) in Eq. (2.22). For other positions of the dividing surface Eq. (2.23) is meaningless. 

Variational transition-state theory (VTST) seeks to find the best lower bound by varying the position of the dividing surface to find the surface that minimizes the rate. 

Fig. la, b. Plot of particle density c profiles vs. distance X vertical to an interface for a two component system A, B to illustrate a possible position of the dividing surface between solid phase and gas phase, (a) Component A in substrate solid phase (b) component B adsorbate. Note that densities are not drawn to scale [73Hir],... 

The total entropy contribution of the surface is obtained by rearrangement of Equation (2.6) and insertion of the entropies per unit volume of a and p. Thus, 5 is determined by the position of the dividing surface. Finally, normalizing by the surface area yields the surface entropy, 5 ... 

In general, the surface energy and the specific surface Helmholtz firee energy are not equal since Eyx depends on the position of the dividing surface. However, if the dividing surface may be chosen such that E = 0 then... 

The position of the dividing surface is generally chosen such that = 0. This choice of position generally means that 0. This is the case in Figure 7.1(b), where the actual amount of B exceeds that represented by the area under the profile defined by the dividing surface. This means that there is an excess of B at the interface, i.e. it is adsorbed ( B > 0). It is clear that, for different concentration profiles, w could be positive, negative or zero. 

Figure 7.3 shows schematic representations of the concentration profiles when oxygen adsorbs on silver. The position of the dividing surface... 

Since the position of the dividing surface is arbitrary, so also are n, IT, etc. they may be positive or negative. In a single-component system we can choose die surface to make n = 0, a dioice whidi is called the equinudar surface. Once this choice is made then all other extensive... 

The thermodynamics of curved surfaces is more subtle than that of planar, and we discuss only the most important case, the spherical surface, for which the two principal radii of curvature are equal. Hie extension of the argument to other curved surfaces is beset with difficulties into which we do not enter. The spherical surface, the bubble or the drop, is the only one that is stable in ffie absence of an external field. The original analysis of Gibbs was clarified and its consequences worked out by Tolman, whose work Koenig extended to multicomponent systems. Buff, Hill, and Kondo describe explicitly how the surface tension depends on the position of the dividing surface to which it is referred, or at which it is calculated. 

Although recognizing that the interfacial region is best considered as an interphase, the alternative mathematical model is to consider the interface as a plane of infinitesimal thickness situated between AA and BB of Fig. 12. This dividing surface can be considered to be positioned so as to give rise to a simplification of (15). Gibbs [183] defined the position of the dividing surface such that the surface excess of constituent 1 is zero, and hence ... 

Here the wavefunction is propagated forward infinitely in time. Then components not located on the product side of the dividing surface are projected out and the remaining wavefunction is propagated backwards in time. It should be noted that the result is invariant with respect to the specific position of the dividing surface. 

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